336 XV NORMED ALGEBRAS AND SPECTRAL THEORY Since w* e J$?2(E) and (u ° H>)* = w* o w* e JS?2(E), it follows that u o y> e JS?2(E) and that, by (11.5.2) and (15.4.8.1), (15.4.8.3) H«ow|l2^IM|-Nl2- Hence j£?2(E) is a self-adjoint two-sided ideal in the Banach algebra with involution J$?(E). In general JS?2(E) is not closed in JS?(E) (Problem 12). For each x e E we can write x = £ £n#n» and IWI2 = X! I£J2- The inter- n n pretation given earlier for a Hilbert-Schmidt operator u in terms of /R shows that the series with general term \£n\ - \\u(an)\\ is convergent, and that (15.4.8.4) \\u(x)\\2g \\x\\2-\\u\\}, so that (15.4.8.5) \\u\\g\\u\\2. This, together with (15.4.8.2), shows that the norm \\n\\2 on the algebra ^f2(E) satisfies the inequality (15.1.1). Also o$?2(E) contains all continuous endomorphisms of finite rank. For if u is of finite rank, then its kernel N = u~~l(G) is closed and of finite codimension in E, hence admits a (finite- dimensional) orthogonal supplement M in E (6.3.1). If (an) is a Hilbert basis of E obtained by taking the union of a Hilbert basis of N and a Hilbert basis of M, then u(a^) = 0 for all but a finite number of indices n; hence u e J5P2(E)- It follows that 3?2(E) has no unit element, for the unit element (if it existed) could only be the identity mapping of E, and this is clearly not a Hilbert- Schmidt operator because E is of infinite dimension. If u 6 o^2(E), if (an) is a Hilbert basis of E and if un is the continuous endomorphism of E such that un(ak) = u(ak) for k ^ n and un(ak) = 0 for k>n, then it is immediate that lim \\u — un\\2 = 0. Hence, in view of H-+QO (15.4.8.5), it follows that u is a compact operator (11.2.10). But there exist compact operators which are not Hilbert-Schmidt operators (Problem 12). It remains to be shown that <^2(E) is complete with respect to the norm \\u\\2- Let (un) be a Cauchy sequence with respect to this norm; then by (15.4.8.5) (un) is also a Cauchy sequence with respect to the norm \\ti\\ on JS?(E), hence converges to an element v e 3?(E) with respect to this norm. On the other hand, there exists a real number $ > 0 such that Iklli- Elem 4 of Section 8.14.)